Tensor Network Renormalization: Supplementary Material

Section A.– RG flow in the space of tensors: the 2Dclassical Ising model.

In this section we provide additional details on the flowthat TNR generates when applied to a tensor networkrepresentation of the partition function of the 2D classicalIsing model on a square lattice, as defined in the maintext.

For all possible values of the temperature T in the Isingmodel, the flow is seen to always end in one of three pos-sible fixed-point tensors: the ordered AZ2 for T < Tc,the critical Acrit for T = Tc, and the disordered Atriv forT > Tc. The ordered and disorder fixed-point tensorscan be expressed exactly with a finite bond dimension,namely χ = 2 and χ = 1, respectively, whereas an exactexpression of the critical fixed-point tensor Acrit is sus-pected to require an infinite bond dimension, and thushere we can only obtain an approximate representation.We emphasize that the non-critical fixed-point tensorsAZ2 and Atriv are equivalent to those previously obtainedby TEFR [1].

For purposes of clarity, instead of following the flowof tensors A(s) we will display instead the flow of theauxiliary tensors B that appear in an intermediate step ofthe coarse-graining transformation, see Fig. 2 in the maintext, as these tensors have smaller bond dimension [2],that is less coefficients, and are thus more easily plotted.However, their behavior under the RG flow is seen to beessentially equivalent to that of tensors A.

Starting with the partition function Z of the 2D clas-sical Ising model at a temperature T , we coarse grainthe corresponding tensor network iteratively using TNR.This generates a sequence of tensors B(s), for multipleRG steps s = 0, 1, 2, . . .. Let [B(s)]ijkl denote the com-ponents of B(s). Here we consider the case where eachindex (i, j, k, and l) has dimension 4. The elements ofthese tensors, reshaped as 16 × 16 matrices [B(s)](ij)(kl)and then normalized such that their singular values sumto unity, are plotted in Fig. A.1.

Below the critical temperature, T = 0.9 TC , we obtaina flow towards a fixed-point tensor BZ2 that has twosignificant elements [BZ2 ]1111 = [BZ2 ]2222 = 0.5, withall other elements zero or arbitrarily small, correspond-ing to the (Z2-)symmetry-breaking phase. At critical-ity, T = TC , the tensors converge to a highly non-trivialfixed point tensor after a small number of RG steps, onewhich appears only slightly changed from the first ten-sor B(0). Note that, due to the truncation error of theTNR scheme, we do not obtain a numerically exact fixedpoint; nonetheless the individual elements of B(2) andB(3) all differ by less than 10−4, while the largest ele-ments of these tensors are of order ∼ 0.1. We thus de-

FIG. A.1. Plots of the elements of tensors [B(s)]ijkl, whenreshaped as 16 × 16 matrices, after s iterations of the TNRcoarse-graining transformation, for several values of s. Darkpixels indicate elements of small magnitude and lighter pix-els indicate elements with larger magnitude. (a) Startingat a sub-critical temperature, T = 0.9 TC , the coarse-grained tensors quickly converge to the Z2 fixed-point ten-sor BZ2 ≡ Btriv ⊕ Btriv. (b) Starting at the critical tem-perature, T = TC , the coarse-grained tensors converge to anon-trivial fixed-point tensor Bcrit. Notice that the differencebetween coarse-grained tensors, |B(s) −B(s+1)|, which is dis-

played with the same color intensity as the plots of |B(s)|, isalready very small (as compared to the magnitude of the ele-ments in the individual tensors) for s = 1. (c) Starting at thesuper-critical temperature, T = 1.1 TC , the coarse-grainedtensors quickly converge to the disordered fixed point Btriv,that has only one non-zero element.

fine Bcrit ≡ B(3) as the approximate critical fixed-pointtensor. The precision with which scale-invariance is ap-proximated over successive RG steps is further examinedin Fig. A.2. Finally, above the critical temperature,T = 1.1 TC , we obtain a trivial fixed-point tensor Btriv.

that has only a single significant element [Btriv]1111 = 1with all other elements zero or arbitrarily small, whichis representative of the infinite temperature, disorderedphase.

2

FIG. A.2. The precision with which TNR approximates ascale-invariant fixed point tensor for the 2D classical Isingmodel at critical temperature Tc is examined by comparingthe difference between tensors produced by successive TNRiterations δ(s) ≡ ∥A(s) −A(s−1)∥, where tensors have been

normalized such that ∥A(s)∥ = 1. For small s (initial RGsteps), the main limitation to realizing scale-invariance ex-actly is physical: the lattice Hamiltonian includes RG ir-relevant terms that break scale-invariance at short-distancescales, but are suppressed at larger distances. On the otherhand, for large s (large number of RG steps) the main obstruc-tion to scale invariance is the numerical truncation errors,which can be thought of as introducing RG relevant terms,effectively throwing us away from criticality and thus scaleinvariance. Indeed, use of a larger bond dimension χ, whichreduces truncation errors, allows TNR to not only achieve amore precise approximation to scale-invariance, but to holdit for more RG steps.

Section B.– Non-critical RG fixed points: TRGversus TNR.

In this section we discuss certain aspects of the flowthat TRG and TNR generate in the space of tensors,which emphasizes one of the main differences betweenthe two approaches. Specifically, we describe a class ofnon-critical fixed-points of the flow generated by TRG,namely those represented by corner double line (CDL)tensors ACDL, and show that TNR coarse-grains suchtensors into a trivial tensor Atriv. We emphasize thatTEFR can also transform a CDL tensor ACDL into atrivial tensor Atriv [1].Fig. B.1(a) contains a graphical representation of a

CDL tensor ACDL, which has components(ACDL

)ijkl

given by(ACDL

)(i1i2)(j1j2)(k1k2)(l1l2)

= δi1j2δj1k2δk1l2δl1i2 (B.1)

where a double index notation i = (i1i2) has the meaningi = i1 + η(i2 − 1). Here the double index runs over val-ues i ∈

{1, 2, . . . , η2

}for integer η, whereas each single

index runs over values i1, i2 ∈ {1, 2, . . . , η}. Notice thata square network formed from such CDL tensors ACDL

FIG. B.1. (a) CDL tensor ACDL of Eq. B.1. (b) Tensornetwork made of CDL tensors, which contains correlationsonly within each plaquette.

contains only short-ranged correlations; specifically onlydegrees of freedom within the same plaquette can be cor-related. We now proceed to demonstrate that this net-work is an exact fixed point of coarse-graining with TRG,which was already described in [1]. Note that it is pos-sible to generalize this construction (and the derivationpresented below) by replacing each delta in Eq. B.1 (e.g.δi1j2) with a generic η × η matrix (e.g. Mi1j2) that con-tains microscopic details. For instance, the fixed pointCDL tensors ACDL(T ) obtained with TRG for the off-critical 2D classical Ising model partition function wouldcontain matricesMi1j2(T ) that depend on the initial tem-perature T . However, for simplicity, here we will onlyconsider the case Mi1j2 = δi1j2 .

In the first step of TRG the tensor ACDL is factorizedinto a pair of three index tensors S,

(ACDL

)ijkl

=

η2∑m=1

SmijSmkl (B.2)

where, through use of the double index notation intro-duced in Eq. B.2, tensors S may be written,

S(m1m2)(l1l2)(i1i2) = δm1l2δl1i2δi1m2 , (B.3)

see also Fig. B.2(a-b). The next step of TRG involvescontracting four S tensors to form an effective tensor A′

for the coarse grained partition function,

A′ijkl =

η2∑m,n,o,p=1

SimpSjnmSkonSlpo (B.4)

3

FIG. B.2. A depiction of the two key steps of the TRG coarse-graining transformation. (a) The first step factorizes the ten-

sor ACDL into a product of two tensors S. (b) Local detail ofthe factorization for CDL tensors, see also Eq. B.2. (c) Thesecond step contracts four tensors S into an effective tensorA′. (d) Local detail of the contraction step, see also Eq. B.5.Notice that the effective tensor A′ is also a CDL tensor (witha 45 degree tilt), indicating that CDL tensors are a fixed pointof the TRG coarse-graining transformation.

where, through use of the explicit form of S given in Eq.B.3, the tensor A′ is computed as,

A′(i1i2)(j1j2)(k1k2)(l1l2) = ηδi1j2δj1k2δk1l2δl1i2 , (B.5)

see also Fig. B.2(c-d). Notice that the effective ten-sor is proportionate to the original CDL tensor, A′

ijkl =

η(ACDL

)ijkl

, where the multiplicative factor of η arises

from the contraction of a ‘loop’ of correlations down to apoint,

η∑m1,n1,o1,p1=1

δm1n1δn1o1δo1p1δp1m1 = η, (B.6)

as depicted in Fig. B.2(d). That the network of CDLtensors is a fixed point of TRG indicates that some ofthe short-range correlations in the tensor network are

preserved during coarse-graining, i.e. that TRG is artifi-cially promoting short-ranged degrees to a larger lengthscale. As a result TRG defines a flow in the space oftensors which is not consistent with what is expectedof a proper RG flow. Indeed, two tensor networks thatonly differ in short-range correlations, as encoded in twodifferent ACDL and ACDL tensors (differing e.g. in thedimension η of each index i1, i2, j1, · · · , or, more gener-ally, on the matrices Mi1j2 discussed above) should flowto the same fixed point, but under TRG they will not.

FIG. B.3. Depiction of the explicit form of the required (a)disentangler u, see also Eq. B.7, (b-c) isometries w and v, seealso Eq. B.8, for coarse-graining the network of CDL tensorsACDL, see Eq. B.1, using TNR.

We next apply the TNR approach to a network ofCDL tensors, as defined in Eq. B.1, demonstrating thatthis network is mapped to a network of trivial tensorsAtriv, with effective bond dimension χ′ = 1, thus fur-ther substantiating our claim that the TNR approachcan properly address all short-ranged correlations at eachRG step. Fig. B.3 depicts the form of disentangler u andisometries v and w that insert into the network of CDLtensors at the first step of TNR, as per Fig. 2,which aredefined as follows. Let us first regard each index of theinitial network as hosting a η2-dimensional complex vec-tor space V ≡ Cη2

, and similarly define a η-dimensionalcomplex vector space V ≡ Cη. Then the disentangleruijkl we use is an isometric mapping u : V⊗ V → V⊗V,with indices i, j ∈ {1, 2, . . . , η} and k, l ∈ {1, 2, . . . , η2},that is defined,

uij(k1k2)(l1l2) =1√η δik1δk2l2δjl1 , (B.7)

where we employ the double index notation, k = (k1, k2)and l = (l1, l2). It is easily verified that u is isometric,u†u = I⊗2, with I as the identity on V. The isometries

4

FIG. B.4. A depiction of the key steps of the TNR coarse-graining transformation in the presence of CDL tensorsACDL.(a) Insertion of disentanglers u and isometries v and w be-

tween two pairs of CDL tensors ACDL results in the elimina-tion of the short-range correlations inside the plaquette thatthose four tensors form. (b)-(c) As a result, the auxiliarytensors B and C only propagate part of the short-range cor-relations, as represented by the existence of two lines. Thesetensors are to be compared with the analogous tensor in TRG,namely A′ in Fig. B.2(d) or ACDL in Fig. B.1(a), which stillcontain four lines. (d) As a result, the new tensor A′, formedby factoring B according to a left-right partition of indicesand C according to an up-down partition of indices, is thetrivial tensor Atriv, which contains no lines and therefore nocorrelations

vijk and wijk are mappings v : V → V ⊗ V that aredefined,

wij(k1k2) = vij(k1k2) =1√η δik1δk2l2δjl1 , (B.8)

where again use double index notation, k = (k1, k2).When inserted into the a 2× 2 block of tensors from thenetwork of CDL tensors, as depicted in Fig. 2(b),thischoice of unitary u and isometries w and v act as exactresolutions of the identity, see Fig. B.4(a). Thus thefirst step of the coarse graining with TNR, as depictedin Fig. 2(a), is also exact. Following the second step ofTNR, Fig. 2(b) one obtains four-index tensors B and C

as depicted in Fig. B.4(b-c), which are computed as,

Bijkl = η4(δilδjk),

Cijkl = η−1(δijδkl), (B.9)

with indices i, j, k, l ∈ {1, 2 . . . η}. The third step of TNR,see Fig. 2(c),involves factoring the tensors B and C intoa products of three index tensors according to a partic-ular partition of indices: B(il)(jk) and C(ij)(kl) respec-tively. However, we see from Eq. B.9 that the tensorsfactor trivially across these partitions, which implies thatthe effective tensor A′ obtained in the final step of theTNR iteration is trivial (i.e. of effective bond dimen-sion χ′ = 1), see also Fig. B.4(d). Thus we have con-firmed that the network of CDL tensors, which containedonly short ranged correlations, can be mapped to a trivialfixed point in a single iteration of TRG, consistent withwhat is expected of a proper RG flow.

Section C.– Critical systems.

In this section we compare the behaviour of TNR andTRG at criticality. TNR produces a critical fixed-pointtensor. In contrast, as first argued by Levin and Navein Ref. [3], TRG does not. Insight into the different be-haviour of TRG and TNR is provided by the scaling of anentropy attached to critical tensors, see Fig. C.3. As ex-pected, this entropy is independent of scale in TNR (sinceTNR produces a critical fixed-point tensor, independentof scale), while it grows roughly linearly with scale inTRG, where no critical fixed-point tensor is reached.

We also examine improved versions of TRG based oncomputing a so-called environment, such as the secondrenormalization group (SRG) [4–6], and show that theyalso fail to produce a critical fixed-point tensor – theentropy grows linearly as in TRG, see Fig. C.5. Fi-nally, we point out that in spite of the fact that Ref.[1] refers to TEFR critical tensors as fixed-point tensors,Ref. [1] presents no evidence that TEFR actually pro-duces a fixed-point tensor at criticality.

A critical system is described by a conformal field the-ory (CFT). We conclude this Section by reviewing thecomputation of scaling dimensions of the CFT by diag-onalizing a transfer matrix of the critical partition func-tion. Any method capable of accurately coarse-grainingthe partition function can be used to extract accurate es-timates of the scaling dimensions, regardless of whetherthe method produces a critical fixed-point tensor. Ac-cordingly, Table I shows that TRG, an improved TRGwith environment, and TEFR indeed produce excellentestimates of the scaling dimensions. Nevertheless, TNRis seen to produce significantly better estimates.

5

C.1 Critical fixed-point tensor in TNR

As discussed in the main text and in Section A, whenapplied to the partition function of the critical Isingmodel, TNR produces a flow {A(0), A(1), · · · } in the spaceof tensors that quickly converges towards a fixed-pointtensor Acrit. By fixed-point tensor Acrit we mean a ten-sor that remains the same, component by component,under further coarse-graining transformations. This isthe case up to small corrections that can be systemati-cally reduced by increasing the bond dimensions χ (seeSection A for quantitative details).Thus, at criticality, TNR explicitly recovers scale in-

variance, as expected of a proper RG transformation.This is analogous to the explicit realization of scale in-variance in critical quantum lattice models obtained pre-viously with the MERA [7, 8] (and, ultimately, closelyrelated to it [9]). In addition to making a key conceptualpoint, realizing scale invariance explicitly in the MERAled to a number of practical results. For instance, it ledto identifying a lattice representation of the scaling op-erators of the theory, and the computation of the scalingdimensions and operator product expansion coefficientsof the underlying CFT [8]. Moreover, it produced anextremely compact representation of the wave-functionof a critical system directly in the thermodynamic limit,avoiding finite size effects. In turn, this resulted in anew generation of approaches both for homogeneous crit-ical systems [8] and for critical systems with impurities,boundaries, or interfaces [10]. Finally, the scale-invariantMERA has become a recurrent toy model as a lattice re-alization of the AdS/CFT correspondence [11]. We thusexpect that the explicit invariance of Acrit, as realized byTNR, will be similarly fruitful.

C.2 Absence of critical fixed-point tensor in TRG

In their TRG paper [3], Levin and Nave already arguedthat TRG should not be expected to produce a fixed-point tensor at criticality. The reason is that a criticaltensor in TRG should be thought of as effectively repre-senting a one-dimensional quantum critical system withan amount of entanglement that scales logarithmic in thesystem size –that is, linearly in the number of coarse-graining step. Thus at criticality the tensor manifestlychanges each time that the system is coarse-grained. Asa matter of fact, the authors referred to this situation asthe breakdown of TRG at criticality (to account for a lin-ear growth of entropy with the number of coarse-graingsteps, the bond dimension χ must grow exponentially,and so the computational cost), and related it to thesimilar critical breakdown of the density matrix renor-malization group [12].In spite of Levin and Nave’s original argument jus-

tifying the breakdown of TRG at criticality, one might

wonder whether there is a local choice of gauge such thatTRG also flows into a fixed-point tensor (up to small nu-merical errors) [13]. This local gauge freedom (calledfield redefinition in [1]), refers to a local change of ba-sis on individual indices of the tensor, implemented byinvertible matrices x and y

Aijkl →∑

i′j′k′l′

(A)i′j′k′l′xii′(x−1)k′kyll′(y

−1)j′j , (C.1)

under which the partition function Z represented by thetensor network remains unchanged. That is, one mightwonder if after a proper choice of local gauge, the criticaltensors A(s) and A(s+1) before and after the TRG coarse-graining step s are (approximately) the same. [We noticethat, in order to simplify the comparison with TNR, hereby one TRG coarse-graining step we actually mean twoTRG coarse-graining steps as originally defined in [3],so that the square lattice is mapped back into a squarelattice with the same orientation].

FIG. C.1. Tensor network representation of matrices M (s)

and Γ(s), as well as the quantities Θ(s)2 and Θ

(s)1 . By construc-

tion, Θ(s)2 and Θ

(s)1 are invariant under local gauge transfor-

mations, as are the eigenvalues of matrices M (s) and Γ(s).

In order to definitely confirm that indeed TRG doesnot produce a critical fixed-point tensor even up to lo-cal gauge transformations, we will study the dependenceof a matrix Γ(s) in the coarse-grainng step s. MatrixΓ(s) is defined in terms of two tensors A(s) accordingto Fig. C.1(b). Γ(s) itself is not invariant under localgauge transformations, but one can build gauge invari-ant quantities out of it. Here we will focus on two suchgauge invariant objects: the spectrum of eigenvalues of

Γ(s), as well as a simple tensor network Θ(s)2 built from

four copies of A(s) and that amounts to

Θ(s)2 ≡ tr

((Γ(s))2

). (C.2)

Fig. C.2 shows the spectrum of eigenvalues {λα(s)}of the normalized density matrix Γ(s)/Θ

(s)1 as a function

of the TRG step s. One can see that the spectrum be-comes flatter as s increases, indicating that more eigen-values become of significant size. [In contrast, the spec-trum remains essentially constant as a function of the

6

FIG. C.2. Spectrum of eigenvalues of the normalized matrix

Γs/Θ(s)1 as a function of the coarse-graining step s, both for

TRG (left) and TNR (right). In TRG, the spectrum becomesflatter with increasing s. In TNR, the spectrum is essentiallyindependent of s for s ≥ 3.

FIG. C.3. Von Neumman entropy S({λα}) and Renyi entropy

R2({λα}) of the eigenvalues of the density matrix Γ(s)/Θ(s)1 .

For TRG, these entropies grow linearly in the coarse-grainingstep s (or logarithmically in the number 2s of quantum spins).For TNR, these entropies are essentially constant for s ≥ 3.

TNR step s]. Fig. C.3 shows the von Neumannm en-tropy S({λα}) ≡ −

∑α λα log(λα) and the Renyi entropy

Rn({λα}) ≡ log(∑

α λnα)/(1− n) of index n = 2

R2 = − log

(∑α

(λα)2

)= − log

(Θ

(s)2

(Θ(s)1 )2

), (C.3)

both of which are invariant under local gauge transfor-mations. These entropies grow linearly in the TRG steps.

The flattening of the spectrum in Fig. C.2, as wellas the steady growth of the entropies S and R2 in Fig.C.3 constitute an unambiguous confirmation that criti-cal TRG tensors A(s) and A(s+1) are not equal up to alocal gauge transformation, not even approximately. Inaddition, these features are robust against increasing thebond dimension χ, and therefore are not an artifact ofthe truncation.

C.3 Use of environment

A significant advance in renormalization methods fortensor networks since the introduction of TRG has beenthe use of the so-called environment to achieve a moreaccurate truncation step. The improved TRG methodproposed in Ref. [4] under the name of “poor-man’sSRG” generates local weights that represent the local en-vironment in a small neighborhood in the tensor network,while the second renormalization group (SRG) proposedin Ref. [5] takes this idea further by using a global envi-ronment that represents the entir tensor network.

Fig. C.4 shows that, for the 2D Ising model at critical-ity, the use of an environment produces a more accuratenumerical estimate of the free energy per site. Specif-ically, the data corresponds to the higher order tensorrenormalization group (HOTRG) proposed in Ref.[6], animprovement on TRG based upon the higher order singu-lar value decomposition (HOSVD) but does not take theenvironment into account; and to the higher order sec-ond renormalization group (HOSRG), which uses bothHOSVD and the global environment. It is also worthpointing out however that TNR, which in its currentimplementation is a purely local update that does nottake into account the environment, produces more accu-rate results than HOSRG for the same value of the bonddimension. [Moreover, the cost in TNR scales only asO(χ6), whereas the cost of HOTRG and HOSRG scalesas O(χ7)].

Based upon the improvement in free energy of HOSRGcompared to HOTRG, one may wonder whether the useof the environment, as in SRG, could potentially re-solve the breakdown of TRG at criticality and even pro-duce a critical fixed-point tensor. We have investigatedthis question numerically by analyzing the RG flow oftensors generated by HOTRG and HOSRG for the 2DIsing model at criticality, as shown in Fig.C.5. Undercoarse-graining with HOSRG, the spectrum of the crit-ical tensors grows increasingly flat with RG step, in asimilar manner as was observed with standard TRG inFig.C.2. The same occurs for HOTRG (not plotted).Likewise, in both HOTRG and HOSRG, the entropy oftensors is seen to increase roughly linearly with RG step,and this growth is essentially robust under increasingthe bond dimension. These results clearly demonstratethat (HO)SRG does not produce a critical fixed-point

7

FIG. C.4. Relative error in the free energy per site δf asa function of bond dimension χ for the 2D Ising model atcriticality, comparing HOTRG, HOSRG and TNR. Throughuse of the environment HOSRG is able to give better accuracythan HOTRG for any given χ, though the accuracy of HOSRGis still less than that of TNR.

tensor. Thus we conclude that the use of environmentin (HO)SRG, while leading to a more accurate coarse-graining transformation for fixed bond dimension over(HO)TRG, does not prevent the computational break-down experienced by TRG at criticality. In particularSRG still fails to produce a proper RG flow and criticalfixed-point tensor.

C.4 The TEFR algorithm

Ref. [1] refers to the tensors obtained with TEFR atcriticality as fixed-point tensors. Accordingly, one mightbe led to conclude that TEFR actually generates a fixed-point tensor (and thus recovers scale invariance) at crit-icality. As demonstrated in Ref. [1], TEFR producesaccurate estimates of the central charge c and scaling di-mensions ∆α for the Ising model, and one might indeedmisinterpret those as being evidence for having generateda critical fixed-point tensor. However, the estimates re-ported in Ref. [1] for c and ∆α are of comparable ac-curacy to those obtained with TRG and “poor-man’sSRG” using the same bond dimension (see Table I inSect. C.5). Since the above methods demonstrably failto produce a fixed-point tensor, the accurate TEFR es-timates provided in Ref. [1] are no evidence that TEFRhas produced a critical fixed-point tensor. In conclusion,Ref. [1] refers to the TEFR critical tensors as fixed-pointtensors, but Ref. [1] provides no evidence that TEFRrealizes a fixed-point tensor at criticality.

FIG. C.5. (left) Spectrum of eigenvalues of the normalized

matrix Γs/Θ(s)1 as a function of the coarse-graining step s for

HOSRG, which becomes flatter with increasing s. (right) VonNeumman entropy S({λα}) of the eigenvalues of the density

matrix Γ(s)/Θ(s)1 , comparing HOTRG, HOSRG and TNR. For

both HOTRG and HOSRG these entropies grow linearly inthe coarse-graining step s (or logarithmically in the number 2s

of quantum spins), indicating that the tensors are not flowingto a scale-invariant fixed point.

C.5 Extraction of scaling dimensions from a transfermatrix

In proposing TRG in Ref. [3], Levin and Nave referto the significant loss of efficiency experienced by themethod at criticality as TRG’s critical break-down. It isimportant to emphasize that, in spite of this significantloss of efficiency at criticality, TRG can still be used toextract universal information about a phase transition,by studying the partition function of a finite system.

The key theoretical reason is that in a finite systemone can observe a realization of the so-called operator-state correspondence in conformal field theories (CFT),which asserts that there is a one-to-one map betweenthe states of the theory and its scaling operators [14].Specifically, the finite size corrections are universal andcontrolled by the spectrum of scaling dimensions of thetheory [15]. This is best known in the context of criticalquantum spin chains, where Cardy’s formula relates thesmallest scaling dimensions ∆α of the conformal theoryto the lowest eigenvalues Eα of the critical Hamiltonianon a periodic chain according to

Eα(L)− Eα(∞) = η2π

L∆α + · · · , (C.4)

where L is the number of spins on the chain, Eα(∞)is the energy in an infinite system, η (independent ofα) depends on the normalization of the Hamiltonian,

8

and where the sub-leading non-universal corrections areO(1/L2) in the absence of marginal operators. Thus, wecan estimate scaling dimensions ∆α by diagonalizing acritical Hamiltonian on a finite periodic chain, as it isoften done using exact diagonalization techniques. In atwo dimensional statistical system, the analogous resultis the observation that even in a finite system, a criticalpartition function Z is organized according to the scalingdimensions of the theory as

Z = eaLxLy

∑α

e−Ly2πLx

(∆α− c12 )+··· ), (C.5)

where a is some non-universal constant, c is the centralcharge of the CFT, and we assumed isotropic couplingsand a square torus made of Lx × Ly sites.As proposed by Gu and Wen [1], if the coarse-grained

tensor A(s) effectively represents the whole partitionfunction of a system made of 2s × 2s sites (and thusLx = Ly = 2s), then we can extract the central chargec and scaling dimensions ∆α from it by e.g. computingthe spectrum of eigenvalues of the matrix M (s) (see Fig.C.1(a)), (

M (s))ik

≡∑j

(A(s)

)ijkj

. (C.6)

Indeed, after removing the non-universal constant a inEq. C.5 [by suitably normalizing the initial tensors A(0)],the largest eigenvalues λα of M (s) are of the form [1]

λα = e−2π(∆α− c12+O( 1

L ). (C.7)

For instance, Table I shows remarkably accurate es-timates of the central charge c and lowest scaling di-mensions ∆α obtained by using TRG and other relatedmethods on a finite lattice made of 1024 spins. These re-sults are a strong indication that TRG accurately coarse-grains the partition function of such a finite system. [Thetable also shows more accurate results obtained withTNR on larger systems using a smaller bond dimension.]In Fig. C.6 we examine the accuracy with which TRG

and TNR reproduce the dominant scaling dimension ofthe Ising model, ∆σ = 1/8, on finite lattices of differ-ent linear size L. Here it can be seen that TRG resultsquickly lose accuracy for larger system sizes, a manifes-tation of the breakdown of TRG at criticality, whilst theresults from TNR maintain a high level of accuracy evenfor very large systems. For instance in comparing theχ = 80 TRG result to the χ = 24 TNR results, whichboth required a roughly equivalent computation time torun on a laptop computer (∼ 3 hours), the TRG calcu-lation gave ∆σ to with 1% accuracy for spins systemswith linear dimension large as L = 211, while the TNRcalculation the same accuracy for systems with linear di-mension large as L = 224: roughly 4000 times largerlinear dimension than TRG.

exact TRG(64) TRG+env(64) TEFR(64) TNR(24)c 0.5 0.49982 0.49988 0.49942 0.50001σ 0.125 0.12498 0.12498 0.12504 0.1250004ϵ 1 1.00055 1.00040 0.99996 1.00009

1.125 1.12615 1.12659 1.12256 1.124921.125 1.12635 1.12659 1.12403 1.125102 2.00243 2.00549 - 1.999222 2.00579 2.00557 - 1.999862 2.00750 2.00566 - 2.000062 2.01061 2.00567 - 2.00168

TABLE I. Exact values and numerical estimates of the centralcharge c and lowest scaling dimensions of the critical Isingmodel. TRG results are obtained using the original Levin andNave’s algorithm [3]. TRG+env results are obtained using animproved TRG method proposed in Ref. [4] under the nameof “poor-man’s SRG”. TEFR results are taken from Ref.[1]. The first three numerical columns use bond dimensionχ ≡ Dcut = 64 and 1024 spins, while the TNR data usesχ = 24 and 262,144 spins.

FIG. C.6. Calculation of the smallest non-zero scaling dimen-sion of 2D classical Ising at criticality, computed using eitherTRG or TNR through diagonalization of the transfer operatoron an effective linear system size of L spins, comparing to theexact result ∆σ = 1/8. For small system sizes, L < 26 spins,finite size effects limit the precision with which ∆σ can becomputed, while for large system sizes truncation errors re-duce the accuracy of the computed value of ∆σ. However, itis seen that the TNR approach maintains accuracy for muchlarger system sizes than does TRG: while TRG with bonddimension χ = 80 gives ∆σ within 1% accuracy up to lin-ear system size L = 211 spins, a TNR with bond dimensionχ = 24, which required a similar computation time as theχ = 80 TRG result, gives 1% accuracy up to L = 224 spins.

In conclusion, TRG is an excellent method to studycritical partition functions on a finite system, providedthat the size of the system that we want to investigateis small enough that the truncation errors are not yetsignificant. In the critical Ising model, sub-leading (non-universal) finite size corrections to the dominant (uni-versal) contributions to the partition function (that is,to the weights e−2π(∆α−c/12) for small ∆α) happen to be

9

very small even for relatively small systems. As a re-sult, one can use TRG to extract remarkably accurateestimates of the central charge c and smallest scaling di-mensions ∆α. This calculation is analogous to extractingscaling dimensions by diagonalizing the Hamiltonian of aquantum critical system on a finite ring and then usingCardy’s formula (Eq. C.4).

However, the ability to extract accurate estimates ofthe central charge and scaling dimensions after applyinga coarse-graining transformation such as TRG on a finitesystem does not imply that one has obtained a criticalfixed-point tensor. Indeed, we have shown above thatTRG does not produce a critical fixed-point tensor, ashad already been anticipated by Levin and Nave [3].

[1] Z.-C. Gu, X.-G.Wen, Phys. Rev. B 80, 155131 (2009).[2] The bond dimension χ of the TNR calculation denotes

the largest bond dimension used. In practice, differentbond dimensions are used at different steps of the TNRcoarse-graining transformation.

[3] M. Levin, C. P. Nave Phys. Rev. Lett. 99, 120601 (2007)[4] H.-H. Zhao, Z.-Y. Xie, Q.-N. Chen, Z.-C. Wei, J. W. Cai,

T. Xiang, Physical Review B 81, 174411 (2010).[5] Z.-Y. Xie, H.-C. Jiang, Q.-N. Chen, Z.-Y. Weng, T. Xi-

ang, Phys. Rev. Lett. 103:160601 (2009).[6] Z.-Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, T.

Xiang, Phys. Rev. B 86, 045139 (2012).[7] G. Vidal Phys. Rev. Lett. 101, 110501 (2008).[8] R. N. C. Pfeifer, G. Evenbly, G. Vidal, Phys. Rev. A

79(4), 040301(R) (2009). G. Evenbly, G. Vidal, QuantumCriticality with the Multi-scale Entanglement Renormal-ization Ansatz, chapter 4 in the book ”Strongly Corre-

lated Systems. Numerical Methods”, edited by A. Avellaand F. Mancini (Springer Series in Solid-State Sciences,Vol. 176 2013)

[9] G. Evenbly, G. Vidal, Tensor network renormaliza-tion yields the multi-scale entanglement renormalizationansatz, arXiv:1502.05385.

[10] G. Evenbly, R. N. C. Pfeifer, V. Pico, S. Iblisdir, L.Tagliacozzo, I. P. McCulloch, G. Vidal, Phys. Rev. B 82,161107(R) (2010). G. Evenbly, G. Vidal, J. Stat. Phys.(2014) 157:931-978.

[11] B. Swingle, Phys. Rev. D 86, 065007 (2012),arXiv:0905.1317. H. Matsueda, M. Ishihara andY. Hashizume, Phys. Rev. D 87, (2013) 066002,arXiv:1208.0206. T. Hartman and J. Maldacena, JHEP05(2013) 014. J. Molina-Vilaplana, J. Prior, Gen.Relativ. Gravit (2014)46:1823. A. Mollabashi, M.Nozaki, S. Ryu, T. Takayanagi, JHEP03(2014)098,arXiv:1311.6095. G. Evenbly, G. Vidal, J Stat Phys(2011) 145:891-918, arXiv:1106.1082. B. Swingle, Con-structing holographic spacetimes using entanglementrenormalization, arXiv:1209.3304 J. Haegeman, T. J.Osborne, H. Verschelde, F. Verstraete, Phys. Rev.Lett. 110, 100402 (2013), arXiv:1102.5524. F. Pastawski,B. Yoshida, D. Harlow, J. Preskill, JHEP 06 (2015)149, arXiv:1503.06237. M. Miyaji, T. Takayanagi, Prog.Theor. Exp. Phys. (2015) 073B03, arXiv:1503.03542. X.-

L. Qi, Exact holographic mapping and emergent space-time geometry, arXiv:1309.6282 N. Bao, C. Cao, S. M.Carroll, A. Chatwin-Davies, N. Hunter-Jones, J. Pollackand G. N. Remmen, Phys. Rev. D 91, 125036 (2015),arXiv:1504.06632.

[12] S. R. White, Phys. Rev. Lett. 69, 2863 (1992). S. R.White, Phys. Rev. B 48, 10345 (1993).

[13] Z.-C. Gu and X.-G. Wen, private communication.[14] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal

Field Theory (Springer, 1997).[15] H. W. J. Blote, J. L. Cardy, M. P. Nightingale, Phys.

Rev. Lett. 56, 724 (1986).